Properties

Label 588.4.a.a
Level $588$
Weight $4$
Character orbit 588.a
Self dual yes
Analytic conductor $34.693$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.6931230834\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{3} - 14 q^{5} + 9 q^{9} + O(q^{10}) \) \( q - 3 q^{3} - 14 q^{5} + 9 q^{9} + 4 q^{11} - 54 q^{13} + 42 q^{15} + 14 q^{17} - 92 q^{19} - 152 q^{23} + 71 q^{25} - 27 q^{27} - 106 q^{29} + 144 q^{31} - 12 q^{33} + 158 q^{37} + 162 q^{39} + 390 q^{41} - 508 q^{43} - 126 q^{45} + 528 q^{47} - 42 q^{51} + 606 q^{53} - 56 q^{55} + 276 q^{57} + 364 q^{59} - 678 q^{61} + 756 q^{65} + 844 q^{67} + 456 q^{69} - 8 q^{71} + 422 q^{73} - 213 q^{75} + 384 q^{79} + 81 q^{81} + 548 q^{83} - 196 q^{85} + 318 q^{87} - 1194 q^{89} - 432 q^{93} + 1288 q^{95} + 1502 q^{97} + 36 q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −3.00000 0 −14.0000 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.4.a.a 1
3.b odd 2 1 1764.4.a.l 1
4.b odd 2 1 2352.4.a.v 1
7.b odd 2 1 84.4.a.b 1
7.c even 3 2 588.4.i.h 2
7.d odd 6 2 588.4.i.a 2
21.c even 2 1 252.4.a.a 1
21.g even 6 2 1764.4.k.n 2
21.h odd 6 2 1764.4.k.c 2
28.d even 2 1 336.4.a.e 1
35.c odd 2 1 2100.4.a.g 1
35.f even 4 2 2100.4.k.g 2
56.e even 2 1 1344.4.a.p 1
56.h odd 2 1 1344.4.a.b 1
84.h odd 2 1 1008.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 7.b odd 2 1
252.4.a.a 1 21.c even 2 1
336.4.a.e 1 28.d even 2 1
588.4.a.a 1 1.a even 1 1 trivial
588.4.i.a 2 7.d odd 6 2
588.4.i.h 2 7.c even 3 2
1008.4.a.d 1 84.h odd 2 1
1344.4.a.b 1 56.h odd 2 1
1344.4.a.p 1 56.e even 2 1
1764.4.a.l 1 3.b odd 2 1
1764.4.k.c 2 21.h odd 6 2
1764.4.k.n 2 21.g even 6 2
2100.4.a.g 1 35.c odd 2 1
2100.4.k.g 2 35.f even 4 2
2352.4.a.v 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 14 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(588))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 14 + T \)
$7$ \( T \)
$11$ \( -4 + T \)
$13$ \( 54 + T \)
$17$ \( -14 + T \)
$19$ \( 92 + T \)
$23$ \( 152 + T \)
$29$ \( 106 + T \)
$31$ \( -144 + T \)
$37$ \( -158 + T \)
$41$ \( -390 + T \)
$43$ \( 508 + T \)
$47$ \( -528 + T \)
$53$ \( -606 + T \)
$59$ \( -364 + T \)
$61$ \( 678 + T \)
$67$ \( -844 + T \)
$71$ \( 8 + T \)
$73$ \( -422 + T \)
$79$ \( -384 + T \)
$83$ \( -548 + T \)
$89$ \( 1194 + T \)
$97$ \( -1502 + T \)
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